70 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA (d) Let E, F be two topological vector spaces and/:E-»F a continuous linear mapping. Then for each bounded subset B of E, the image set/(B) is bounded in F. 7, (a) Let E be a (real or complex) Hausdorff topological vector space. Show that, if there exists a bounded neighborhood V of 0 in E (Problem 6), the sets w~1V (n an integer >0) form a fundamental system of neighborhoods of 0 in E, and consequently E is metrizable. If furthermore E is locally convex, then its topology can be defined by a single norm. (b) Let E be a metrizable locally convex space whose topology is defined by an increasing sequence (pn) of seminorms. Show that the topology of E can be defined by a single norm if and only if there exists an integer N such that, for all n ^> N, there exists a real number kn ^ 0 for which pn <ji knp^. (c) Let E be the real vector space of indefinitely differentiable real-valued functions on the interval I — [0,1 ] in R. For each n J> 0 and each/e E, put />„(/)= sup (sup|D*/(r)|) 04*<nVreI / (where D°/=/). Show that each pn is a norm on E and that the topology defined by the sequence of norms (pn) cannot be defined by a single norm (cf. (17.1)). (d) Let E be a Hausdorff locally convex space whose topology is defined by a se- quence of seminorms but cannot be defined by a single norm. If d is a translation- invariant distance defining the topology of E (12.9.2) show that every set in E which is bounded in the sense of the topological vector space structure of E is bounded with respect to the distance d, but there exist subsets of E which are bounded with respect to dbut are unbounded with respect to the topological vector space structure of E. 8. Let E be a metrizable topological vector space. (a) Show that every balanced subset of E which absorbs all sequences in E which converge to 0 is a neighborhood of 0 in E. Deduce that, if # is a linear mapping of E into a topological vector space F, and w transforms every sequence converging to 0 in E into a bounded sequence in F, then w is continuous. (b) Let (Bn) be any sequence of bounded subsets of E. Show that there exists a sequence (An) of nonzero scalars such that the union of the sets AnBn is bounded. 9. Let E be a topological vector space. A set S3 of bounded subsets of E is called a, funda- mental system of bounded subsets if every bounded set in E is contained in a set belonging to 83. Show that, if E is a metrizable locally convex space whose topology cannot be defined by a single norm, no fundamental system of bounded subsets of E is denumerable (use Problem 8). 10, Let (EM)n2ii be a sequence of normed spaces such that Ex is not separable. Let F be a vector subspace of E =O E« > such that PrB(F) = En for each n. Show that there exists n a real number 8 > 0 and a bounded sequence (xm)mzi in F such that whenever / ^j. (Remark that, for each index n and eachnondenumerable subset A of F, there exists a nondenumerable subset B of A such that prn(B) is bounded in En .) Deduce that a Frechet space is separable if every bounded subset is relatively compact. 11. A subset A of a real or complex vector space E is convex if, whenever x and y belong to A and 0 <; A <; 1, we have AJC + (1 — X)y e A (Section 8.5, Problem 9). It followsounded if and only if pra(B) is bounded